Arguably, the most natural and powerful framework to study boolean modal logics in a uniform and systematic way, is the theory of Boolean Coalgebraic Logics (henceforth BCL, see e.g. [1]}). In its `abstract' ([2]}) presentation, it is parametrised by an endofunctor \(L:\mathbf {BA}\rightarrow \mathbf {BA}\)  which builds modal algebras of modal terms over a boolean structure, an endofunctor \(T:\mathbf {Set}\rightarrow \mathbf {Set}\)  which builds the transition structures over which the modal terms are to be interpreted, and a natural transformation \(\delta : L\mathsf {P}\rightarrow \mathsf {P}T^{\mathrm {op}}\)  (where \(\mathsf {P}:\mathbf {Set}^{\mathrm {op}}\rightarrow \mathbf {BA}\)  is the powerset functor) which implements the interpretation by associating sets of acceptable successors states to each modal term over a predicate (see [3]}, [4]}, [5]}, [6]}). This data, and the dual adjunction between \(\mathbf {Set}\)  and \(\mathbf {BA}\) , is traditionally summarized in the following diagram
\({\mathbf {BA}@/^1pc/[rr]^{\mathsf {S}} @(dl,ul)^{L}& \perp &\mathbf {Set}^{\mathrm {op}}@/^1pc/[ll]^{\mathsf {P}}@(dr,ur)_{T^{\mathrm {op}}} & & & \delta : L\mathsf {P}\Rightarrow \mathsf {P}T^{\mathrm {op}}}\) 
